Abstract. We study single-flip dynamics in sets of three-dimensional rhombus tilings with fixed polyhedral boundaries. This dynamics is likely to be slowed down by so-called 'cycles': such structures arise when tilings are encoded via the 'partition-on-tiling' method and are susceptible to break connectivity by flips or at least ergodicity, because they locally suppress a significant amount of flip degrees of freedom. We first address the so-far open question of the connectivity of tiling sets by elementary flips. We prove exactly that sets of tilings of codimension 1 and 2 are connected for any dimension and tiling size. For higher codimension tilings of dimension 3, the answer depends on the precise choice of the edge orientations, which is a non-trivial issue. In most cases, we can prove connectivity despite the existence of cycles. In the few remaining cases, among which one is the icosahedral symmetry, the question remains open. We also study numerically flip-assisted diffusion to explore the possible effects of the previously mentioned cycles. Cycles do not seem to slow down significantly the dynamics, at least as far as self-diffusion is concerned.

PACS numbers: 61.44.Br, 02.10.−v, 02.40.−k, 45.05.+x

Print publication: Issue 1 (7 January 2005)
Received 13 July 2004, in final form 20 October 2004
Published 8 December 2004